Question #80383

A torsional pendulum consists of a disc of moment of inertia $I \text{ (kg m}^2\text{)}$ and a suspension wire of torsional constant $C \text{ (N m rad}^{-1}\text{)}$. A closed-loop coil of $N$ turns and area $A \text{ (m}^2\text{)}$ is rigidly attached to the disc and placed in a uniform radial magnetic field $B \text{ (T)}$. The coil has an initial electrical resistance $R_0 \text{ (}\Omega\text{)}$ at temperature $T_0 \text{ (K)}$, a temperature coefficient of resistance $\alpha \text{ (K}^{-1}\text{)}$, and a heat capacity $H \text{ (J K}^{-1}\text{)}$. The pendulum is given an initial angular displacement $\theta_0 \text{ (rad)}$ and released. The damping is purely electromagnetic. Assuming all dissipated mechanical energy is converted into heat within the coil with no heat loss to the surroundings, what is the final quality factor $Q_f$ of the system after the oscillations have completely damped out? (Given the initial quality factor is $Q_0$)